COMETARY ACTIVITY BEYOND THE PLANETS

It has long been known that some comets are active far beyond the range of heliocentric distances over which water ice can sublimate, the latter corresponding roughly to the region interior to Jupiter’s orbit (semimajor axis 5 AU). The interpretation of activity in distant comets is complicated. Ices more volatile than water (e.g. CO, CO₂) may be involved, as might the exothermic crystallization of amorphous water ice, at least at distances $r_{H}\lesssim$ 12 AU (Jewitt, 2009; Guilbert-Lepoutre et al., 2012). In comets observed after perihelion, the origin of distant activity is further complicated by the slow conduction into the interior of heat acquired at perihelion, resulting in the delayed activation of sub-surface volatiles and driving distant outbursts (e.g. Prialnik & Bar-Nun (1992)).

Significantly, in long-period comets that are active at large distances while still inbound to perihelion, the conduction of perihelion heat can play no role. The long-period comet C/2017 K2 (PANSTARRS), which was discovered active pre-perihelion at heliocentric distance $r_{H}=16$ AU, falls into this category. Observations reveal a spherical coma of large ($\sim$0.1 - 1 mm) size grains, slowly ejected (speeds $\sim$ 4 m s^-1) close to steady-state from a nucleus no more than 9 km in radius, Jewitt et al. (2017, 2019a). Coma was present in prediscovery observations at 23.8 AU, a linear extrapolation of photometry indicates that activity began at $r_{H}\sim$ 26 AU, (Jewitt et al., 2019a) while more detailed considerations indicate that C/2017 K2 was active even at $\sim$35 AU (Jewitt et al., 2021). Comet C/2010 U3 (Boattini) likewise possessed a coma at 25.8 AU pre-perihelion (Hui et al., 2019), while C/2014 B1 (Schwartz) was also active far beyond the water sublimation zone with a coma of large, slowly-moving grains, but observed only starting at 11.9 AU, pre-perihelion (Jewitt et al., 2019b). Recently, comet C/2014 UN271 (Bernardinelli-Bernstein) was reported to be active at 23.8 AU (Farnham, 2021) and may have been active at much larger distances.

Free sublimation of exposed super-volatile ices provides an initially attractive but ultimately unsatisfactory explanation for the distant activity. Gas drag forces produced by free-sublimation of CO in equilibrium with sunlight at distances $>$ 15 AU are orders of magnitude too small to eject 100 $\mu$m sized particles against the gravity of the nucleus (Jewitt et al., 2019a). Neither could smaller particles be ejected owing to inter-particle (van der Waals) cohesive forces naturally between grains. Taken at face value, this “cohesion bottleneck” prevents the ejection of particles of any size in distant comets, as noted by Gundlach et al. (2015). However, sublimation in a confined space, for example beneath a porous refractory mantle, offers the possibility of building the pressure to values much higher than reached in free sublimation to space. Fulle et al. (2020) have suggested that sublimation occurs within centimeter-sized “pebbles” whose small but finite strength allows gas pressure to build up to values sufficient to eject dust by gas drag. Activity driven by the crystallization of amorphous water ice has also been proposed as a mechanism but, at $\geq$ 15 AU, temperatures are too low for this process to occur.

We here explore solutions to the bottleneck problem involving the build-up of pressure by the sublimation of sub-surface ice, taking careful account of the processes by which heat and gas transport occur. In so doing, we find that cometary activity is possible at extraordinarily large distances, even far beyond the planetary region of the solar system. We present the model in Section (2) and the results in Section (3).

The physical model we present is a refinement of Whipple’s (1950) icy conglomerate model, enhanced to include the presence of a non-volatile, low conductivity porous mantle (see Figure 1). In active comets, such a “rubble mantle” is produced by fallback of suborbital particles and by volatile loss from the strongly heated surface. In inactive comets, for example those in the Oort cloud and Kuiper belt storage reservoirs, an irradiation mantle can grow as pure ices are rendered non-volatile by prolonged exposure to cosmic rays (Kaiser & Roessler (1997), Abplanalp, Frigge, & Kaiser (2019), Gautier et al. (2019)). In our model, we assume that a non-volatile mantle of variable thickness and porosity overlies CO ice.

In equilibrium with sunlight, the energy balance averaged over the surface of a spherical, isothermal nucleus is written as

The first term on the left of Equation (1), in which $F_{\odot}$ is the solar constant, $r_{H}$ is the heliocentric distance in AU and $A_{B}$ is the bond albedo, represents the absorbed solar power. The second term on the left, only important at the largest heliocentric distances, accounts for heating of the nucleus from other external sources. These include the cosmic microwave radiation background and the integrated light of the stars, as represented by an effective temperature $T_{OC}$ = 10 K. The emissivity of the nucleus is $\epsilon_{IR}$ while $\sigma$ is the Stefan-Boltzmann constant. On the right, the first term is the flux of radiation from the nucleus surface at temperature, $T$, while the second term is the conducted flux. There is no contribution from sublimation since the surface layers are assumed to be volatile-free.

The conductivity in Equation (1), $k(T)$, is written as in Mendis & Brin (1977)

As shown schematically in Figure (1), the ice is not on the surface as in Whipple’s model but hidden under a mantle. Energy propagated through the mantle to deeper layers can trigger the sublimation of buried ices. At the ice-mantle interface, the conducted energy flux sublimates ice at rate

Here, the sublimation is entirely controlled by the conducted heat; a non-equilibrium process. The generated gas is also subject to another non-equilibrium process caused by the pressure gradient. The heat is driven down the temperature gradient towards the ice front, $B$ in Figure (1), while the gas is driven up towards the surface of the comet, $A$, by the pressure gradient established in the porous medium. In this simple comet model the reverse flow of molecules is not favored, so condensation is somewhat unlikely and is neglected for simplicity. The sublimation flux that has been calculated in Watson & al. (1961) and in Delsemme, & Miller (1971) approximates the sublimation flux in our case too. Otherwise, we should model the condensation process and calculate the net flux

Practically, $f_{CO}$ is an upper limit to the sublimated flux because of the possibility of back-flow and sticking of molecules reflected from particles above the ice front. This is the necessary Neumann boundary condition. At all levels above the ice front, we calculate the net flux, $J_{CO}$, formed as molecules diffuse through the porous medium.

In equation (4), $P(T)$ is the equilibrium pressure of the $CO$ gas over the $CO$ ice as function of temperature. $P(T)$ contains valuable information about both the gas and the ice structure. We consider sublimation and condensation as facets of a single process which is a first order phase transition phenomenon. It is worthwhile to invoke the chemical reaction formalism to address it. Sublimation is considered as the direct process while condensation is the reverse process; analogous to the direct and reverse arrows between reactants and products in chemical reaction theory. Usually, experiments are conducted in closed thermodynamic equilibrium to characterize the chemical reaction and find the equilibrium constant called $K(T)$. We found that for pure ice, the temperature dependence of $P(T)$ varies just like $K(T)$ in the language of chemical reactions. The resulting pressure relation resembles the Arrhenius equation (5). Physically, the similarity arises because in a chemical reaction we consider the breakage of bonds within molecules while, in sublimation, the breaking bonds are those between molecules. For this reason, the activation energies $E_{i}$ in the sublimation case are small compared to the energies involved in chemical reactions.

We fitted the sublimation data from Fray & Schmitt (2009) using

Our model is governed by two coupled partial differential equations, one for heat transport

and one for mass transport

In Equation (6) $\rho$ is the density and $k(T)$ is the conductivity at temperature, $T$, while $t$ is the time and $x$ is the depth in the nucleus, and $c_{p}(T)$ (see equations 8 and 9) is the heat capacity of the mantle. In Equation (7), $D_{K}=\frac{r_{p}}{3}V_{th}$ is the Knudsen diffusion coefficient, $V_{th}=\sqrt{\frac{8k_{B}T}{\pi m}}$ is the thermal speed, $\kappa=\frac{r_{p}^{2}}{32}$ the permeability of the porous medium (Cussler, 2009), and $\mu=\frac{\alpha}{\pi^{3/2}}\frac{\sqrt{k_{B}mT}}{\sigma^{2}}$ is the dynamic viscosity. Other quantities are $k_{B}$, the Boltzmann constant, $\alpha$, a numerical constant on the order of 1, $T$, the temperature, $\sigma$ the CO molecule’s collision diameter and $m$ the CO molecular mass, $p$, the CO gas pressure $p(n,T)=nk_{B}T$, $n$ is the molecular density of CO; $r_{p}$ the pore radius, $\epsilon$ is the porosity.

The equations are coupled by the transport coefficients. Water ice acts as a mineral at the low temperatures found in the outer solar system, so we neglect the H₂O gas flux and focus only on the CO flux using a simplified version of the model described in Bouziani & Fanale (1998). Gas produced by sublimation passes through the porous mantle towards the comet surface. Figure (3) illustrates the various modeled mechanisms of gas transport inside the porous mantle. This model takes into account both microscopic and macroscopic behavior and also treats the interaction between the gas and the walls. It provides a continuous physical modeling of the coupling between mechanisms - from discontinuous collisions to the collective macroscopic behavior of continuous matter - and thus for the prediction of the mass flow rate and pressure distribution with reasonable accuracy over the widest range of gas pressures. The model depends only on molecular data and fundamental parameters such as temperature, composition and total pressure. This approach allows us to address the thermal and dynamical coupling of comets in a continuous and physical way. Table (1) summarizes parameters used in the numerical simulations.

We assume that the dust mantle is composed of a mixture of refractory minerals and water ice. To estimate the heat capacity of the ice, we use the heat capacity $c_{p}$ (J kg^-1 K^-1)deduced from the data provided in Shulman (2004).

For the refractory part of the mantle we use the data for Enstatite found in Krupka & al (1985) and we find that the Shulman equation gives an excellent representation of the data at both high and low temperatures;

We solved Equations (6 & 7) as a function of time, following the motion of comet C/2017 K2 upon its approach to the Sun from large heliocentric distances. For this purpose, we used the barycentric orbital elements from Królikowska & Dybczyński (2018) while noting that the current Sun-centered orbit from NASA’s Horizons ephemeris software¹¹1http://ssd.jpl.nasa.gov/horizons.cgi is slightly hyperbolic (eccentricity $e$ = 1.0004, vs. $e$ = 0.9999 from Królikowska & Dybczyński (2018)). This difference between the barycentric and Sun-centered elements is negligible for our purposes. The model solves the heat and gas transport equations in a coupled manner. As it approaches the Sun, the surface temperature of C/2017 K2 increases from the limiting interstellar value of 10 K to higher temperatures, creating a temperature gradient within the mantle, and driving conduction towards the buried ice layer. Heating of the ice triggers sublimation, producing a pressure gradient and resulting in a diffusive flow of gas towards the comet’s surface, eventually leading to escape and the expulsion of embedded dust particles through gas drag. The downward conduction of heat and the upward transport of sublimated gas molecules are characterized by distinct and different timescales, each dependent on the thermal and micro-physical properties of the material, as we will discuss in Section (3.2). The boundary condition (Equation 1) is time-dependent because the orbital position is a function of time.

We present a detailed gas-physics model of sublimation of cometary CO ice buried beneath a permeable, refractory dust mantle. The model is solved as a function of time and distance along the orbit of the distantly active long-period comet C/2017 K2.

1. 1.

   Our main result is that we find an unexpected local peak in CO production at very large heliocentric distances ($r_{H}\approx~{}150$ AU), caused by the build up of pressure beneath a permeable mantle of modest thickness.

2. 2.

   This peak, whose magnitude is a function of several mantle microphysical parameters, corresponds to the Mendis point, where the buried CO ice front transitions between distinct temperature regimes.

3. 3.

   While modeled on C/2017 K2 as a specific case, our conclusions are general. Comets entering the solar system from Oort cloud and interstellar distances may become active far beyond the planets, at heliocentric distances $r_{h}\sim$ 150 AU. Observational attempts to detect such ultra-distant activity are encouraged.

Here, $k$ is the thermal conductivity of the material, $\rho$ and $c_{p}$ are respectively the density and heat capacity. Quantity $\tau$ is the heating timescale.

The conducted heat flux into the cold core is $F_{C}=k(dT/dx)\approx k\Delta T/l$, where $\Delta T=T_{CO}-T_{C}$ is the temperature difference between the top and bottom of the heated layer of thickness $l$. Then

For comparison, the energy flux used by sublimation is

Conduction into the cold core is smaller than sublimation when $F_{C}<F_{sub}$, leading to